General Theory of Algebraic EquationsEtienne Bézout

Translated from the French by Eric Feron

This book provides the first English translation of Bezout's masterpiece, the General Theory of Algebraic Equations. It follows, by almost two hundred years, the English translation of his famous mathematics textbooks. Here, Bézout presents his approach to solving systems of polynomial equations in several variables and in great detail. He introduces the revolutionary notion of the "polynomial multiplier," which greatly simplifies the problem of variable elimination by reducing it to a system of linear equations. The major result presented in this work, now known as "Bézout's theorem," is stated as follows: "The degree of the final equation resulting from an arbitrary number of complete equations containing the same number of unknowns and with arbitrary degrees is equal to the product of the exponents of the degrees of these equations."

The book offers large numbers of results and insights about conditions for polynomials to share a common factor, or to share a common root. It also provides a state-of-the-art analysis of the theories of integration and differentiation of functions in the late eighteenth century, as well as one of the first uses of determinants to solve systems of linear equations. Polynomial multiplier methods have become, today, one of the most promising approaches to solving complex systems of polynomial equations or inequalities, and this translation offers a valuable historic perspective on this active research field.

Etienne Bézout (1730-1783) is credited with the invention of the determinant (named Bézoutian by Sylvester) as well as several key innovations to solve simultaneous polynomial equations in many unknowns. By the time of his death, he was a member of the French Academy of Sciences and the Examiner of the Guards of the Navy and of the Corps of Artillery. Eric Feron Dutton/Ducoffe Professor of Aerospace Engineering at Georgia Institute of Technology, and Visiting Professor of Aerospace Engineering at Massachusetts Institute of Technology, where he is affiliated with the Laboratory for Information and Decision Systems and the Operations Research Center. He is also an Adviser to the French Academy of Technologies. His interests span numerical analysis, optimization, systems analysis, and their applications to aerospace engineering.

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Reviews

"This is not a book to be taken to the office, but to be left at home, and to be read on weekend, as a romance. We already know the plot, but here we meet all the characters, major and minor."--Cicero Fernandes de Carvalho, Mathematical Reviews

"Bézout's classic General Theory of Algebraic Equations is . . . an immortal evergreen of astonishing actual relevance. . . . [I]ts first English translation is utmost welcome and appropriate, and a great gain for the international mathematical community, both today and in the future."--Werner Kleinert, Zentralblatt MATH

Table of Contents

Definitions and preliminary notions 1
About the way to determine the differences of quantities 3
A general and fundamental remark 7
Reductions that may apply to the general rule
to differentiate quantities when several differentiations must be made. 8
Remarks about the differences of decreasing quantities 9
About certain quantities that must be differentiated through a simpler process than that resulting from the general rule 10
About sums of quantities 10
About sums of quantities whose factors grow arithmetically 11
Remarks 11
About sums of rational quantities with no variable divider 12

Book One

Section I

About complete polynomials and complete equations 15
About the number of terms in complete polynomials 16
Problem I: Compute the value of N(u . . . n)T 16
About the number of terms of a complete polynomial that can be divided by certain monomials composed of one or more of the unknowns present in this polynomial 17
Problem II 17
Problem III 19
Remark 20
Initial considerations about computing the degree of
the final equation resulting from an arbitrary number of complete equations with the same number of unknowns 21
Determination of the degree of the final equation resulting from an arbitrary number of complete equations containing the same number of unknowns 22
Remarks 24

Section II

About incomplete polynomials and first-order incomplete equations 26
About incomplete polynomials and incomplete equations in which each unknown does not exceed a given degree for each unknown. And where the unknowns, combined two-by-two, three-by-three, four-by-four etc., all reach the total dimension of the polynomial or the equation 28
Problem IV 28
Problem V 29
Problem VI 32
Problem VII: We ask for the degree of the final equation resulting from an arbitrary number n of equations of the form (u a . . . n)t = 0 in the same number of unknowns 32
Remark 34
About the sum of some quantities necessary to determine the number of terms of various types of incomplete polynomials 35
Problem VIII 35
Problem IX 36
Problem X 36

Problem XI 37
About incomplete polynomials, and incomplete equations, in which two of the unknowns (the same in each polynomial or equation) share the following characteristics:
(1) The degree of each of these unknowns does not exceed a given number (different or the same for each unknown);
(2) These two unknowns, taken together, do not exceed a given dimension;
(3) The other unknowns do not exceed a given degree (different or the same for each), but, when combined groups of two or three among themselves as well as with the first two, they reach all possible dimensions until that of the polynomial or the equation 38
Problem XII 39
Problem XIII 40
Problem XIV 41
Problem XV 42

Problem XVI 42
About incomplete polynomials and equations, in which three of the unknowns satisfy the following characteristics:
(1) The degree of each unknown does not exceed a given value, different or the same for each;
(2) The combination of two unknowns does not exceed a given dimension, different or the same for each combination of two of these three unknowns;
(3) The combination of the three unknowns does not exceed a given dimension.
We further assume that the degrees of the n - 3 other unknowns do not exceed given values; we also assume that the combination of two, three, four, etc. of these variables among themselves or with the first three reaches